Scheme Theory

26 flashcards · Other · by ceyhun elmacıoğlu

1. GGG group, GnG_nGn​ descending filtration. GGG is Hausdorff iff what?

   ⋂nGn={0}\bigcap_{n}G_n = \{0\}⋂n​Gn​={0}

2. GGG and G′G'G′ groups, GnG_nGn​ and Gn′G'_nGn′​ descending filtrations. The topologies are same iff what?

   For every nnn , there is an mmm such that Gm′⊆GnG'_m\subseteq G_nGm′​⊆Gn​ and vice versa

3. GGG group, GnG_nGn​ descending filtration. A sequence (xn)(x_n)(xn​) is Cauchy if what?

   For every nnn , there is an m0m_0m0​ such that xm−xm0∈Gnx_m-x_{m_0}\in G_nxm​−xm0​​∈Gn​ for every m≥m0m\geq m_0m≥m0​

4. Inverse limit of an inverse system (An)(A_n)(An​)

5. Universal property of inverse limit

6. Construction of completion

7. Universal property of completion

8. Isomorphism related to completion

9. Completion of the polynomial ring A[T1,…,Tn]A[T_1,\dots,T_n]A[T1​,…,Tn​]

   Formal power series A[[T1,…,Tn]]A[[T_1,\dots,T_n]]A[[T1​,…,Tn​]]

10. Let MMM be a finitely generated AAA - module. Then M⊗AA^undefinedM^M\otimes_A \hat{A} \xrightarrow[]{} \hat{M}M⊗A​A^ ​M^ is surjective

11. Formal completion A^\hat{A}A^ of a Noetherian ring AAA is

    Noetherian flat over over AAA

12. Inverse Limit = Limit Direct Limit = Colimit

13. AAA Noetherian local, m\mathfrak{m}m its maximal ideal, m\mathfrak{m}m -adic completion A^\hat{A}A^ . What do we know?

    A/mn≅A^/mnA^A/\mathfrak{m}^n \cong \hat{A}/\mathfrak{m}^n\hat{A}A/mn≅A^/mnA^ A^\hat{A}A^ is a local ring with maximal ideal mA^\mathfrak{m}\hat{A}mA^

14. What is V(I)V(I)V(I) ?

    Set of prime ideals containing III as a subset

15. What is D(f)D(f)D(f) ?

    Set of prime ideals not containing fff

16. V(I)∪V(J)=V(I)\cup V(J) =V(I)∪V(J)=

    V(I∩J)V(I\cap J)V(I∩J)

17. ⋂λ∈ΛV(Iλ)=\bigcap\limits_{\lambda\in\Lambda}V(I_\lambda)=λ∈Λ⋂​V(Iλ​)=

    V(∑λ∈ΛIλ)V\Big( \sum_{\lambda\in \Lambda} I_\lambda \Big)V(∑λ∈Λ​Iλ​)

18. V(A)=V(A) =V(A)= V(0)=V(0) = V(0)=

    ∅\emptyset∅ SpecA\text{Spec}ASpecA

19. A singleton {p}\{\mathfrak{p}\}{p} is closed in SpecA\text{Spec} A SpecA iff

    p\mathfrak{p}p is a maximal ideal of AAA

20. True/False: V(I)=V(I)V(I) = V(\sqrt{I})V(I)=V(I ​)

    True

21. What is I\sqrt{I}I ​ equal to?

    ⋂p∈V(I)p=⋂p⊇Ip\bigcap\limits_{\mathfrak{p}\in V(I)} \mathfrak{p} = \bigcap\limits_{\mathfrak{p}\supseteq I} \mathfrak{p}p∈V(I)⋂​p=p⊇I⋂​p

22. V(I)⊆V(J)V(I)\subseteq V(J)V(I)⊆V(J) iff

    J⊆IJ\subseteq \sqrt{I}J⊆I ​

23. Given φ:AundefinedB\varphi: A\xrightarrow[]{} Bφ:A ​B , how is SpecBundefinedSpecA\text{Spec}B \xrightarrow[]{} \text{Spec}A SpecB ​SpecA defined?

    Spec(p)=φ−1(p)\text{Spec}(\mathfrak{p}) = \varphi^{-1}(\mathfrak{p})Spec(p)=φ−1(p)

24. True/False: Spec\text{Spec}Spec is continuous

    True

25. If φ:AundefinedB\varphi: A \xrightarrow[]{} Bφ:A ​B is surjective, what is SpecB\text{Spec}BSpecB homeomorphic to?

    SpecB≅\text{Spec}B \congSpecB≅ V(ker⁡φ)V(\ker\varphi)V(kerφ)

26. If φ:AundefinedS−1A\varphi: A \xrightarrow[]{} S^{-1}Aφ:A ​S−1A is the localization map, what is Spec(S−1A)\text{Spec}(S^{-1}A)Spec(S−1A) homeomorphic to?

    {p∈A ∣ p∩S=∅}\{\mathfrak{p}\in \text{A} \ \big\vert \ \mathfrak{p}\cap S = \emptyset \}{p∈A ∣ ∣​ p∩S=∅}